Completions and Formal Power Series Extensions

Abstract

This concluding chapter is concerned mainly with the behavior of the divisor class group under base change to the formal power series ring. It has been seen that Cl(A) → Cl(A[[T]]) is not always a bijection, even when A is factorial. Using the good functorial properties of the Picard group and its relation to the divisor class group, Danilov has studied the class of rings for which the homomorphism is a bijection. A fundamental concept is the Picard group of an open subscheme of Spec A. For example, the set of points in Spec A at which an (divisorial) ideal is invertible is open, and on this subscheme, the class of the ideal belongs to the Picard group. Using this simple property, it is possible to show that there is a natural splitting of the injection Cl(A) → Cl(A[[T]]).